Inelastic material behavior of polymers – Experimental characterization,formulation and implementation of a material model

In this contribution an experimental procedure based on displacement controlled tensile tests at different rates of loading, relaxation experiments and deformation controlled loading and unloading processes with intermediate relaxations to experimentally characterize and classify the nonlinear, inelastic mechanical behavior of polymers is presented. These experiments provide data for a structured approach to parameter identification. In line with the experiments, a small strain uniaxial viscoplastic material model is derived, subsequently generalized to multiaxial loadings and implemented into a finite element program. The combination of the experimental procedure and the proposed material model is then used to characterize and model the mechanical behavior of the thermoplastic polypropylene. After the identification of the necessary material parameters, stress–strain curves have been computed for different uni- and multiaxial loadings and are compared with experimental results.     

On the numerical handling of fractional viscoelastic material models in a FE analysis

The paper presents the finite element (FE) implementation of linear and nonlinear fractional viscoelasticity models. To this end, a short introduction on fractional calculus is given. In addition to the fractional operators, this includes analytical and numerical solution schemes for selected fractional integral and differential equations. The presented rheological model is based on state of the art approaches and has been adopted to model the strain rate dependent material behavior of polymers. To this end, two approaches with constant and overstress dependent viscous properties resulting in linear and nonlinear evolution equations are discussed. The uniaxial constitutive relations are generalized to the multiaxial case and processed to be implemented in a FE code. The model behavior of both approaches is demonstrated and compared for selected uniaxial and multiaxial load cases.     

高聚物材料动态模量测量与分析

高聚物材料具有良好的阻尼结构特性,因此在减振降噪中发挥很大作用。高聚物材料的模量是复数,并且随着系统的频率和环境温度呈非线性变化。将某高聚物材料在动态热机械分析仪上进行动态力学行为实验,并对实验结果采用分数导数维模型进行拟合。结果表明,分数导数维Maxwell模型可以同时精确的拟合高聚物材料的存储模量和损耗模量随频率变化的曲线,而且其形式简单,统一,计算所需参数较少,研究结果为实际应用提供了参考依据。     

Finite element formulation of viscoelastic sandwich beams using fractional derivative operators

This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler–Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.     

Micro-Macro Analysis of Viscoelastic Unidirectional Laminated Composite Plates Using DR Method

The Dynamic Relaxation (DR) technique together with finite difference discritization is used to study the bending behavior of Mindlin composite plate including geometric nonlinearity. The overall behavior of the unidirectional composite is obtained from a three-dimensional (3D) micromechanical model, in any combination of normal and shear loading conditions, based on the assumptions of Simplified Unit Cell Method (SUCM). The composite system consists of nonlinear viscoelastic matrix reinforced by transversely isotropic elastic fibers. A recursive formulation for the hereditary integral of the Schapery viscoelastic constitutive equation in multiaxial stress state is used to model the nonlinear viscoelastic matrix material in the material level. The creep tests data is used for verification of the predicted response of the current approach. Under uniform lateral pressure, the laminated plate deformation with clamped and hinged edged constraints is predicted for various time steps.     

Development of a stress-mode sensitive viscoelastic constitutive relationship for asphalt concrete: experimental and numerical modeling

Asphalt binder is responsible for the thermo-viscoelastic mechanical behavior of asphalt concrete. Upon application of pure compressive stress to an asphalt concrete specimen, the stress is transferred by mechanisms such as aggregate interlock and the adhesion/cohesion properties of asphalt mastic. In the pure tensile stress mode, aggregate interlock plays a limited role in stress transfer, and the mastic phase plays the dominant role through its adhesive/cohesive and viscoelastic properties. Under actual combined loading patterns, any coordinate direction may experience different stress modes; therefore, the mechanical behavior is not the same in the different directions and the asphalt specimen behaves as an anisotropic material. The present study developed an anisotropic nonlinear viscoelastic constitutive relationship that is sensitive to the tension/compression stress mode by extending Schapery’s nonlinear viscoelastic model. The proposed constitutive relationship was implemented in Abaqus using a user material (UMAT) subroutine in an implicit scheme. Uniaxial compression and indirect tension (IDT) testing were used to characterize the viscoelastic properties of the bituminous materials and to calibrate and validate the proposed constitutive relationship. Compressive and tensile creep compliances were calculated using uniaxial compression, as well as IDT test results, for different creep-recovery loading patterns at intermediate temperature. The results showed that both tensile creep compliance and its rate were greater than those of compression. The calculated deflections based on these IDT test simulations were compared with experimental measurements and were deemed acceptable. This suggests that the proposed viscoelastic constitutive relationship correctly demonstrates the viscoelastic response and is more accurate for analysis of asphalt concrete in the laboratory or in situ.     

The use of stress sweep tests for asphalt mixtures nonlinear viscoelastic and fatigue damage responses identification

Asphaltic materials are known to present a behavior that can be approximated by the theory of viscoelasticity. For these materials it is essential to characterize fatigue damage. An important aspect therein is the separation between nonlinear viscoelastic and fatigue damage responses. This is a complex issue, since both nonlinearity and damage have a similar effect on the overall material mechanical behavior, i.e. decrease in the stiffness and increase in the phase angle. This paper presents an experimental and a mathematical procedure to separate the nonlinear viscoelastic from the fatigue damage response for asphaltic materials. Stress sweep tests were used to characterize a hot mixture asphalt at nine conditions (three temperatures and three frequencies). Once all strain values were obtained in a stress controlled sweep test, a statistical analysis was used to find the maximum stress that can be applied to the material without invoking the damage response. The results showed that the transition stress value is directly associated with material properties, the stiffness being an important factor in this result. Consequently, stress, temperature and frequency determine together the mechanical response of the material (linear or nonlinear viscoelastic, fatigue damage and/or plastic deformation). Results from this study can be associated with other fatigue damage approaches in order to better select the stress or strain amplitude that should be used in fatigue tests, and to eliminate the amount of energy that is dissipated in the nonlinear viscoelastic region.     

A viscoelastic continuum damage model and its application to uniaxial behavior of asphalt concrete

An existing viscoelastic constitutive model which accounts for the effects of rate-dependent damage growth is described and applied successfully to characterize the uniaxial stress, constant strain rate behavior of asphalt concrete. The special case of an elastic continuum damage model with multiaxial loading, which is based upon thermodynamics of irreversible processes with internal state variables, is first reviewed and then it is shown how this model has been extended to a corresponding viscoelastic damage model through the use of an elastic-viscoelastic correspondence principle. The general mathematical model is next specialized to uniaxial loading. A rate-type evolution law, similar in form to a crack growth law for a viscoelastic medium, is adopted for describing the damage growth within the body. Results from laboratory tests of uniaxial specimens under axial tension at different strain rates are then shown to be consistent with the theory. The discussion of data analysis describes the specific procedure used here to obtain the material parameters in the constitutive model for uniaxial loading and how the method may be generalized for multiaxial loading.     

基于非线性黏弹性本构模型的轮胎滚动和生热

轮胎的滚动阻力和自生热是造成轮胎失效的原因之一,轮胎内部能量的产生主要取决于轮胎中橡胶材料的黏弹性能量耗散。文中基于超弹性模型和并行流变模型(PRF)描述了橡胶材料的非线性黏弹性响应特征,提出了一种预测实心轮胎温度分布和滚动阻力的方法。首先将线性黏弹性Prony级数转化为PRF模型的初始参数,并利用Isight软件根据多应变工况加载得到的应力松弛测试数据来校准材料参数,然后采用显式-热力耦合分析方法分析基于Prony级数和PRF模型的实心轮胎滚动过程的差异。结果表明,Prony级数无法描述橡胶材料的非线性行为,在显式动力学下计算的轮胎生热结果为0;PRF模型可以表征橡胶材料的非线性行为,并且计算的轮胎模型在0.3s内温度上升了0.14℃。     

Relaxation behavior of neat and particulate filled polypropylene in uniaxial and multiaxial compression

The recently developed confined compression test was used to measure the viscoelastic bulk and shear relaxation moduli of neat, glass bead and talc filled polypropylene. In this paper further modifications of the test are introduced and a criterion for the assessment of the quality of experimental data is suggested. As expected, shear as well as the bulk relaxation moduli were found to increase with the addition of particles. In order to determine the pressure sensitivity of the material, unconfined compression tests were also performed and compared with the confined tests through interconversion of the measured moduli. In agreement with earlier results on other polymers, it turned out that the relaxation response is significantly retarded at higher confinement levels. It is shown that the effect of filler particles on the long-term behavior depends on the specific uniaxial or multiaxial stress state. Poisson’s ratio was calculated by interconversion from the bulk and shear relaxation modulus; these results show that with a single test in the confined configuration, a complete viscoelastic characterization of the material can be obtained.     

Modeling short- and long-term time-dependent nonlinear behavior of polyethylene

A new methodology for developing macromechanical constitutive formulations for time-dependent materials is presented in this article. In particular, two phenomenological constitutive models for polymer materials are illustrated, describing time-dependent and nonlinear mechanical behavior. In this new approach, short-term creep test data are used for modeling both short-term and long-term responses. The differential form of a model is used to simulate typical nonlinear viscoelastic polymeric behavior using a combination of springs and dashpots. Unified plasticity theory is then used to develop the second model, which is a nonlinear viscoplastic one. Least squares fitting is applied for the determination of material parameters for both models, based on experimental results. Due to practical constraints, experimental data are usually available for short-term time-frames. In the presented proposed formulation, the material parameters determined from short-term testing are used to obtain material parameter relationships for predicting the long-term material response. This is done by extending short-term information for longer time frames. Finally, theoretical and experimental results of tensile tests on polyethylene subjected to various load levels and test times are compared and discussed. Very good agreement of the modeling results with experimental data shows that the developed formulation provides a flexible and reliable framework for predicting load responses of polymers.     

The viscoelastic response of cement paste to three-dimensional loading

The majority of the viscoelastic constitutive data for cement paste or concrete found in the literature deal exclusively with uniaxial loading. To predict the isotropic response of concrete or cement paste under multiaxial loading or multiaxial prescribed deformation, it is necessary to have knowledge of at least two viscoelastic constitutive properties. In the past, the typical treatment of three-dimensional modeling of concrete viscoelasticity has involved the assumption of a time-independent viscoelastic Poisson ratio. However, the experimental evidence supporting this simplification is inconclusive. In this study, experiments were performed on hardened cement paste that allowed the simultaneous measurement of both the dilatational and shear compliances, allowing the full three-dimensional characterization of the constitutive response. It was found that the dilatational compliance leveled off after several days for three of four mixtures tested. In these three materials, the Poisson’s ratio was found to be an increasing function of time. Prediction of the measured uniaxial compliance using the measured bulk and shear compliances indicated that the confined compressive test used in this research may cause changes in the material which affect the measured dilatational compliance, and therefore the calculated viscoelastic Poisson ratio.     

Modeling multiaxial impact behavior of a glassy polymer

In many applications of polymers, impact performance is a primary concern. Impact tests experimentally performed on molding prototypes yield useful data for a particular structural and impact loading case. But, it is generally not practical in terms of time and cost to experimentally characterize the effects of a wide range of design variables. A successful numerical model for impact deformation and failure of polymers can provide convenient and useful guidelines on product design and therefore decrease the disadvantages that arise from purely experimental trial and error. Since the specimen geometry and loading mode for multiaxial impact test provides a close correlation with practical impact conditions and can conveniently provide experimental data, the first step of validating a numerical model is to simulate this type of test. In this paper, we create a finite element analysis model using ABAQUS/Explicit to simulate the deformation and failure of a glassy ABS (acrylonitrile-butadiene-styrene) polymer in the standard ASTM D3763 multiaxial impact test. Since polymers often exhibit different behavior in uniaxial tensile and compression tests, the uniaxial compression or tensile tests are generally not representative of the three-dimensional deformation behavior under impact loading. A hydrostatic pressure effect (controlled by the parameter γ) is used to generalize a previously developed constitutive model ("DSGZ" model) so that it can describe the entire range of deformation behavior of polymers under any monotonic loading modes. The generalized DSGZ model and a failure criterion are incorporated in the FEA model as a user material subroutine. The phenomenon of thermomechanical coupling during plastic deformation is considered in the analysis. Impact load vs. displacement and impact energy vs. displacement curves from FEA simulation are compared with experimental data. The results show good agreement. Finally, equivalent stress, strain, strain rate and temperature distributions in the polymer disk are presented. Electronic Publication     

竖向集中力作用下分数导数型半无限体粘弹性地基变形分析

研究成果表明地基土体具有粘弹性性质,在长期荷载作用下会产生蠕变变形。为了准确预测地基变形,必须建立精确的土体本构模型。分数导数粘弹性模型具有精确度高,确定模型所需的实验参数少,应用范围广的特点。针对半无限体粘弹性地基,采用分数导数粘弹性本构模型模拟土的力学行为,运用弹性-粘弹性对应原理分析了粘弹性地基的变形。研究发现,采用分数导数粘弹性本构模型得到的地基沉降量较经典粘弹性模型要小,经典粘弹性模型不能很好地反映集中力对周围地基沉降的影响以及地基的蠕变性质。     

非线性黏弹复合材料有效性质

提出了一个细观力学模型 , 用于预测非线性黏弹聚合物基复合材料的有效性质。该方法利用广义割线模量方法对单积分型热力学本构进行线性化 , 并运用 Laplace变换技术将黏性问题转化为弹性问题。利用热力学本构拟合高密度聚乙烯的实验数据 , 得到基体的材料参数。 利用该模型计算了玻璃微珠填充高密度聚乙烯复合材料(GB/HDPE)在恒应变率下的应力应变关系 , 计算结果与文献实验结果吻合较好。数值计算结果表明 GB/HDPE复合材料表现出明显的非线性力学行为。 该细观力学模型可以很好地预测复合材料非线性黏弹性性质。      

基于分数阶黏弹性模型的木塑复合材料蠕变/回复性能分析

木塑复合材料(WPC)是一种木质纤维增强聚合物的新型环保复合材料,为分析WPC在非恒定荷载下的变形行为,进行结构的长期变形设计,对WPC的蠕变/回复变形进行计算分析。采用叠加原理对比分析既有蠕变计算模型对WPC蠕变/回复的整体预测效果。结果表明,现有模型均不能良好预测其蠕变/回复行为。采用基于分数阶微积分的黏弹性模型对其蠕变/回复行为进行预测,提出一种双参数法的修正分数阶黏弹性模型。通过与已有实测数据对比表明,该模型能够准确反映WPC的静态黏弹性行为。结合实验数据,给出了不同WPC蠕变/回复模型的参数取值。      

Estimation of biaxial tensile and compression behavior of polypropylene using molecular dynamics simulation

The polymeric materials in general exhibit strong time–temperature dependence and viscoelastic behavior. The time–temperature superposition principle is typically used to estimate the long-term viscoelastic behavior. In addition, Mises criterion and Tresca criterion have been proposed to estimate the yield or failure stresses in a multiaxial stress state and Christensen failure criterion can be applied in the case of different tensile and compressive strengths. In this study, using molecular dynamics method, uniaxial and biaxial tensile and compression test simulations were performed for polypropylene at various strain rates and temperatures. It was observed that the compressive fracture stresses were higher than the tensile fracture stresses. In addition, the fracture stress was high at a low temperature and high strain rate and these fracture stresses are in good agreement with Christensen failure criterion curves. Furthermore, the long-term viscoelastic behavior can almost be predicted from the short-term viscoelastic behaviors at three different temperatures using time–temperature superposition principle. But, the simulations at a wide range of temperatures is important to predict the more accurate long-term viscoelastic behavior.     

正弦型黏弹性拱的非线性动力学行为研究

利用分数导数的本构关系建立了粘弹性拱的控制方程,采用Galerkin方法简化了拱的数学模型。提出一种求解含分数算子的非线性方程的数值方法,并利用该方法对控制方程进行求解。考察载荷参数、材料参数对拱动力响应的影响。运用非线性动力学中各种经典的分析方法,如时程曲线、功率谱、相图、庞加莱截面等,判别并揭示了粘弹性拱的丰富的动力学行为。     

On the use of a spectrum-based model for linear viscoelastic materials

A new spectrum-based model for describing the behavior of time-dependent materials is presented. In this paper, unlike most prior modeling techniques, the time-dependent response of viscoelastic materials is not expressed through the use of series. Instead, certain criteria have been imposed to select a spectrum function that has the potential of describing a wide range of material behavior. Another consequence of choosing the spectrum function of the type used in this paper is to have a few closed form analytic solutions in the theory of linear viscoelasticity. The Laplace transform technique is used to obtain the necessary formulae for viscoelastic Lame' functions, relaxation and bulk moduli, creep bulk and shear compliance, as well as Poisson's ratio. By using the Elastic–Viscoelastic Correspondence Principle (EVCP), material constants appearing in the proposed model are obtained by comparing the experimental data with the solution of the integral equation for a simple tensile test. The resulting viscoelastic functions describe the material properties which can then be used to express the behavior of a material in other loading configurations. The model's potential is demonstrated and its limitations are discussed.     

FEM Implementation of a Three-Dimensional Viscoelastic Constitutive Model for Particulate Composites with Damage Growth

In order to analyze viscoelastic behavior of particulatecomposites with growing damage, an existing three dimensionalviscoelastic continuum damage model developed originally for solidpropellant is generalized for wider use in a Finite Element model(FEM). This equation allows for damage induced anisotropy (localtransverse isotropy). The constitutive equation is modified hereto account for the change of the material continuously froma compressible, undamaged isotropic state into the damagedanisotropic state. A fully viscoelastic time-dependentimplementation of the constitutive equation in a FEM is achievedthat allows for future extension of the FEM to simultaneously takeviscoplasticity into account. The computational results arecompared to experimental results for uniaxial and multiaxialstress states in displacement-driven experiments for solidpropellant. The multiaxial stress experiments used wide stripswith a center hole. The model predicts the experimental load andlocal strain response up to, and slightly beyond, the peak load,very well. The algorithm is shown by example to be stable far pastthe peak load.